When does $$\text{sup}_{x\in A} \text{sup}_{y\in B}f(x,y) = \text{sup}_{y\in B}\text{sup}_{x\in A} f(x,y)$$ fail?
Further can we also have then that $$\text{sup}_{x \in A, y\in B} f(x,y)$$ is greater than both of the expressions above?
(By principle of iterated supremum, we need at least that $\text{sup}_{x \in A, y\in B} f(x,y)=\infty$).